Finite Group Databasebeta

← #13166228

Subgroups — #13166228

Order 1728.

202 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3792C211yesmaximal, characteristic
2#11724242C212yesmaximal, complemented
3#10593274—31nomaximal
4C22×C3×C3⋊Dic3C411yescharacteristic
5C3×#3792—22no—
6C2×C3×#622C411yescharacteristic
7C2×C3×#622C2211yescharacteristic
8C3⋊#3771—22no—
9C2×#3792D311yescharacteristic
10#64218—32no—
11C23×C33C811yescharacteristic
12C2×C3×C3⋊Dic3—21no—
13C2×C3×C3⋊Dic3—21no—
14C2×C3×C3⋊Dic3C811yescharacteristic
15C2×C3×C3⋊Dic3C2×C411yescharacteristic
16C3×#622—21no—
17C3×#622D412yes—
18#4553—91nomaximal
19C22×C3⋊Dic3Dic311yescharacteristic
20C22×C3×Dic3—121no—
21#3792—22no—
22C2×#622Dic311yescharacteristic
23C2×#622C2×D311yescharacteristic
24#3771—62no—
25C22×C33—21no—
26C22×C33—41no—
27C22×C33C2×C811yescharacteristic
28C3×C3⋊Dic3—21no—
29C3×C3⋊Dic3C22⋊C411yescharacteristic
30C3×C3⋊Dic3M1611yescharacteristic
31C2×C3×M16—91no—
32#812—92no—
33C23×C32—41no—
34C23×C32—81no—
35C23×C32C3⋊C811yescharacteristic
36C2×C3⋊Dic3—21no—
37C2×C3⋊Dic3—21no—
38C2×C3⋊Dic3C3⋊C811yescharacteristic
39C2×C3⋊Dic3C2×Dic311yescharacteristic
40C2×C3×Dic3—121no—
41C2×C3×Dic3—121no—
42C2×C3×Dic3—122no—
43C2×C3×Dic3—122no—
44#622—21no—
45#622#6712yes—
46#349—271noSylow
47C2×C33—21no—
48C2×C33—41no—
49C2×C33C22⋊C811yescharacteristic
50C22×C3×C4—91no—
51C22×Dic3—121no—
52C3×M16—182no—
53C2×C3×C8—91no—
54C2×C3×C8—91no—
55C3⋊C16—182no—
56C22×C32—21no—
57C22×C32—42no—
58C22×C32—41no—
59C22×C32—44no—
60C22×C32—41no—
61C22×C32—84no—
62C22×C32—82no—
63C22×C32—81no—
64C22×C32C2×C3⋊C811yescharacteristic
65C3⋊Dic3—21no—
66C3⋊Dic3#20811yescharacteristic
67C3⋊Dic3#21711yescharacteristic
68C3×Dic3—121no—
69C3×Dic3—121no—
70C3×Dic3—122no—
71C2×M16—91no—
72M32—272no—
73C33#34911yescharacteristic, complemented, Sylow
74C23×C3—41no—
75C23×C3—81no—
76C23×C3AGL(1,𝔽9)11yescharacteristic
77C2×C3×C4—91no—
78C2×C3×C4—91no—
79C2×C3×C4—181no—
80C2×C3×C4—181no—
81C2×Dic3—121no—
82C2×Dic3—121no—
83C2×Dic3—122no—
84C2×Dic3—122no—
85C3×C8—92no—
86C3×C8—181no—
87C2×C32—21no—
88C2×C32—42no—
89C2×C32—41no—
90C2×C32—44no—
91C2×C32—41no—
92C2×C32—84no—
93C2×C32—82no—
94C2×C32—81no—
95C2×C32#83011yescharacteristic
96C22×C4—91no—
97M16—182no—
98C2×C8—91no—
99C2×C8—91no—
100C16—542no—
101C22×C3—21no—
102C22×C3—42no—
103C22×C3—41no—
104C22×C3—44no—
105C22×C3—41no—
106C22×C3—82no—
107C22×C3—84no—
108C22×C3—81no—
109C22×C3C2×AGL(1,𝔽9)11yescharacteristic
110C3×C4—91no—
111C3×C4—91no—
112C3×C4—181no—
113Dic3—121no—
114Dic3—121no—
115Dic3—122no—
116C32—41no—
117C32—81no—
118C32#455311yescharacteristic, complemented
119C23C3⋊AGL(1,𝔽9)11yescharacteristic
120C2×C4—91no—
121C2×C4—91no—
122C2×C4—181no—
123C2×C4—181no—
124C8—92no—
125C8—181no—
126C2×C3—21no—
127C2×C3—42no—
128C2×C3—44no—
129C2×C3—41no—
130C2×C3—41no—
131C2×C3—82no—
132C2×C3—84no—
133C2×C3—81no—
134C2×C3C22⋊AGL(1,𝔽9)11yescharacteristic
135C22—21no—
136C22—41no—
137C22C2×C3⋊AGL(1,𝔽9)11yescharacteristic
138C4—91no—
139C4—91no—
140C4—181no—
141C3—41no—
142C3—81no—
143C3#1059327411yescharacteristic, complemented
144C2—21no—
145C2—41no—
146C2#1172424311yescharacteristic
subgroupG/H
centerC2#11724243
commutatorC22×C33C2×C8
FrattiniC22C2×C3⋊AGL(1,𝔽9)
Fitting, prime radicalC23×C33C8
socleC2×C33C22⋊C8
cosocleC2×C3×#622C22
hypercenterC23C3⋊AGL(1,𝔽9)
hypocenterC33#349
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C23

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11724242C2semidirect
73. C33#349semidirect
118. C32#4553semidirect
143. C3#10593274semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#37922
#117242422
#105932743
#45539